AND MAXIMAL DIVISIBLE RESIDUATED LATTICE OF QUOTIENTS
JUSTIN PARALESCU
Communicated by the former editorial board
The aim of this paper is to define the notions of divisible residuated lattice of multipliers, divisible residuated lattice of fractions and maximal divisible resid- uated lattice of quotients for a divisible residuated lattice. The results obtained are generalizations of the ones obtained forBL-algebras in [4]. In the last part of this paper, we prove the existence of the maximal divisible residuated lattice of quotients for a divisible residuated lattice (Theorem 4.3) and we give explicit descriptions of this divisible residuated lattice for some classes of divisible resid- uated lattices.
AMS 2010 Subject Classification: 06D35, 03G25.
Key words: M V-algebra, BL-algebra, residuated lattice, multiplier, divisible residuated lattice, divisible residuated lattice of fractions, maximal divisible residuated lattice of quotients.
1. INTRODUCTION
The origin of residuated lattices is in Mathematical Logic without con- traction. They have been investigated by Krull [18], Dilworth [13], Ward and Dilworth [29], Ward [28], Balbes and Dwinger [1] and Pavelka [21].
MV-algebras are known to be special residuated lattices with a proper additive operation ⊕. M V-algebras fulfill a double negation law x∗∗=x,too.
For a residuated latticeLwe denote byM V(L) the set of all elementsx∗ =x→ 0 with x ∈ L. (M V(L),⊕,∗,0) is an M V− algebra iff for all x, y∈ L,(x∗ → y∗) → y∗ = (y∗ → x∗) → x∗ (see [25]), where for x, y ∈ M V(L), x⊕y = x∗ → y . In particular, for any semi-divisible residuated lattice L, the subset (M V(L),⊕,∗,0) is anM V-algebra (see [25]).
The concept of maximal lattice of quotients for a distributive lattice was defined by J. Schmid in [23, 24] taking as a guide-line the construction of a complete ring of quotients by partial morphisms introduced by G. Findlay and J. Lambek (see [19], p. 36). For the case of BL-algebras see [4]. The central
MATH. REPORTS16(66),3(2014), 381–411
role in this construction is played by the concept of multiplier (defined for a distributive lattice by W.H. Cornish in [11, 12]). The paper is organized as follows. In Section 2, we recall the basic definitions and we present the M V- center of a divisible residuated lattice (defined by Turunen and Mertanen in [25]). This is a very important construction, which associates an M V-algebra with every divisible residuated lattice. In this way, many properties can be transfered from M V-algebras to divisible residuated lattices and backwards.
In Section 3, we define the notion ofmultiplier for a divisible residuated lattice;
also, we put in evidence many results which we need in the rest of the paper.
In Section 4, we define the notions of divisible residuated lattice of fractions and maximal divisible residuated lattice of quotients for a divisible residuated lattice. In the last part of this paper, we prove the existence of the maximal divisible residuated lattice of quotients for a divisible residuated lattice (Theo- rem 4.3) and we give explicit descriptions of this divisible residuated lattice for some classes of divisible residuated lattices (local divisible residuated lattices, chains, and Boolean algebras).
2. DEFINITIONS AND FIRST PROPERTIES
2.1. DEFINITIONS
We review the basic definitions related to residuated lattices. Also, we present the M V-center of a semi-divisible residuated lattice, defined by Tu- runen and Mertanen in [25].
Definition 2.1. Aresiduated lattice[2, 26] is an algebra (L,∨,∧,,→,0,1) of type (2,2,2,2,0,0) equipped with an order≤ satisfying the following:
(LR1) (L,∨,∧,0,1) is a bounded lattice, whose order is≤;
(LR2) (L,,1) is a commutative monoid;
(LR3) and → form an adjoint pair, i.e. a ≤ x → y iff ax ≤ y for all a, x, y∈L.
With the notations in the definition above, (L,,1,≤) becomes a com- mutative ordered monoid (see Theorem 2.2, (c11)).
The relation between the pair of operationsand→expressed by (LR3) is a particular case of the law of residuation [2]. The class RL of residuated lattices is equational.
For any residuated latticeLand everyx∈L, we shall denotex∗ =x→0 and x∗∗= (x∗)∗.
Example 2.1 ([26]). Let p be a fixed non-zero natural number and I = [0,1] the real unit interval. If we define for x, y ∈ I, xy = (max{0, xp+
yp−1})1/pand x→y= min{1,(1−xp+yp)1/p},then (I,max,min,,→,0,1) becomes a residuated lattice calledgeneralized Lukasiewicz structure.
For p = 1 we obtain the notion of Lukasiewicz structure (x y = max{0, x+y−1}, x→y= min{1,1−x+y}).
Example 2.2 ([26]). If we consider on I = [0,1], to be the usual mul- tiplication of real numbers and for x, y ∈ I, x → y = 1 if x ≤ y and y/x otherwise, then (I,max,min,,→,0,1) is a residuated lattice; moreover, it is a product algebra (see below).
Example 2.3 ([26]).If (B,∨,∧,0,0,1) is a Boolean algebra, then if we define for everyx, y∈B, xy=x∧y andx→y=x0∨y,then (B,∨,∧,,→ ,0,1) becomes a residuated lattice.
In a residuated lattice L, we consider the following identities, for every x, y∈L:
(BL1) x(x→y) =x∧y (divisibility);
(BL2) (x→y)∨(y→x) = 1 (prelinearity);
(BL∗1) [x∗(x∗ →y∗)]∗ = (x∗∧y∗)∗ (semi-divisibility).
Definition 2.2.The residuated lattice L is called:
(i) divisible ifL satisfies (BL1);
(ii) M T L-algebra ifLsatisfies (BL2);
(iii) BL-algebra ifLsatisfies (BL1) and (BL2) (that is,Lis a divisibleM T L- algebra);
(iv) semi-divisible ifL satisfies (BL∗1).
A product algebra is a BL-algebra L which fulfills the following condi- tions:
• for all x∈L,x∧x∗ = 0;
• for all x, y, z∈L,z∗∗((xz)→(yz))≤x→y.
We denote by RL ( RLd,MT L, BL, RLsd) the class of residuated lat- tices (divisible residuated lattices,M T L-algebras, BL-algebras, semi-divisible residuated lattices, respectively).
Proposition 2.1 ([15]). For a residuated lattice L, the following condi- tions are equivalent:
(i) L∈ RLd;
(ii) For every x, y∈L with x≤y there exists z∈L such that x=yz;
(iii) For every x, y, z∈L, x→(y∧z) = (x→y)[(x∧y)→z].
Remark 2.1.Following Theorem 3.1 from [17], every divisible residuated lattice is distributive.
We recall [8, 26] that an M V-algebra is an algebra (M,⊕,∗,0) of type (2,1,0) such that:
(M V1) (M,⊕,0) is a commutative monoid;
(M V2) x∗∗=x,for every x∈M;
(M V3) (x → y) → y = (y → x) → x, for every x, y ∈ M (where x → y = x∗⊕y).
We denote by MV the class ofM V-algebras.
Remark 2.2 ([26]).
1. It is not hard to see that an equivalent presentation ofM V-algebras can be given asBL-algebras plus condition (M V2).
2. Let (L,∨,∧,,→,0,1) be a residuated lattice. If forx, y∈L we denote x⊕y = x∗ → y, then (L,⊕,∗,0) is an M V-algebra iff (x → y) → y = (y →x)→x,for everyx, y∈L.
Remark 2.3. Lukasiewicz structures and Boolean algebras areBL-algebras;
not every residuated lattice is a BL-algebra (see [26], p. 16).
In what follows by L we denote the universe of a residuated lattice; for x∈L and a natural numbern, we definex∗=x→0, x∗∗= (x∗)∗, x0 = 1 and xn=xn−1x forn≥1.
In residuated lattices we have the following rules of calculus:
Theorem 2.2 ([5, 6, 9, 15, 26]). Let x, x1, x2, y, y1, y2, z ∈ L. Then we have:
(c1) 1→x=x, x→1 = 1,0→x= 1, x0 = 0;
(c2) x≤y iff x→y = 1;
(c3) x≤y→x, x≤(x→y)→y,((x→y)→y)→y=x→y;
(c4) x→y≤(z→x)→(z→y);
(c5) x→y≤(y→z)→(x→z);
(c6) (x→y)→(x→z)≤x→(y→z);
(c7) x→(y∧z) = (x→y)∧(x→z),(y∧z)→x≥(y→x)∨(z→x);
(c8) (x∨y)→z= (x→z)∧(y→z);
(c9) x≤y implies z→x≤z→y and y→z≤x→z;
(c10) x1 →y1 ≤(y2 →x2)→[(y1 →y2)→(x1 →x2)];
(c11) x≤y implies zx≤zy;
(c12) x(x→y)≤x∧y, x≤y→(xy);
(c13) x(x→(xy)) =xy;
(c14) x→(y→z) = (xy)→z=y→(x→z);
(c15) x → y ≤ (xz) → (yz) and (x1 → y1)(x2 → y2) ≤ (x1x2) → (y1y2);
(c16) (x1→y1)(x2→y2)≤(x1∨x2)→(y1∨y2);
(c17) (x1→y1)(x2→y2)≤(x1∧x2)→(y1∧y2);
(c18) x∨y= 1⇒xy=x∧y;
(c19) x∨(yz)≥(x∨y)(x∨z),so, xm∨yn≥(x∨y)mn for any natural numbers m, n;
(c20) x(y→z)≤y→(xz)≤(xy)→(xz);
(c21) x(y∨z) = (xy)∨(xz);
(c22) (x→y)n≤xn→yn for every natural number n;
(c23) (x∧y)n≤xn∧yn, for every natural number n;
(c24) (x∨y)n≥xn∨yn, for every natural number n.
Theorem 2.3 ([5, 6, 9, 15, 26]). If x, y∈L, then : (c25) 1∗ = 0,0∗= 1;
(c26) x→y≤y∗ →x∗; (c27) x≤y⇒y∗ ≤x∗;
(c28) xx∗= 0 andxy= 0 iff x≤y∗; (c29) x≤x∗∗, x∗∗≤x∗ →x, x∗∗∗=x∗; (c30) (xy)∗=x→y∗ =y→x∗; (c31) (x∨y)∗ =x∗∧y∗;
(c32) x∗∗→y∗∗=y∗→x∗=x→y∗∗; (c33) (x→y∗∗)∗∗=x→y∗∗;
(c34) (x→y)∗∗≤x∗∗→y∗∗;
(c35) x∗∗y∗∗≤(xy)∗∗,so (x∗∗)n≤(xn)∗∗ for every natural number n;
(c36) x∗y∗ ≤(xy)∗.
Corollary 2.4 ([7]). If Lis a divisible residuated lattice, then for every x, y∈L we have:
(c37) (x∗∗→x)∗ = 0;
(c38) (x→y)∗∗=x∗∗→y∗∗;
(c39) (xy)∗∗=x∗∗(x∗∗∧y∗)∗,(x∧y)∗∗=x∗∗∧y∗∗; (c40) y∗≤x⇒x→(xy)∗∗=y∗∗;
(c41) x(y∧z) = (xy)∧(xz), x∧(y∨z) = (x∧y)∨(x∧z).
2.2. MV-CENTER OF A DIVISIBLE RESIDUATED LATTICE For a residuated latticeLwe consider the subset ofL, M V(L) ={x∗:x∈ L} ={x ∈L:x∗∗ =x}. M V(L) is non-void, since 0 = 1∗, 1 = 0∗ ∈M V(L).
Moreover,M V(L) is closed with respect to the operations→(because by (c30), x∗ →y∗= (x∗y)∗ ∈M V(L)), ∧(by (c31)) and to the operation∗.
For x, y∈L we definex⊕y=x∗ →y.
Theorem 2.5 ([25]). (M V(L),⊕,∗,0)is anM V-algebra iff for allx, y∈ L,
(x∗ →y∗)→y∗ = (y∗ →x∗)→x∗.
Theorem 2.6 ([25]). A residuated latticeL is semi-divisible iff (x∗→y∗)→y∗ = (y∗ →x∗)→x∗
holds in L.
Corollary 2.7 ([25]). In any semi-divisible residuated lattice L, (M V(L),⊕,∗,0)is an M V-algebra (called the M V-center of L).
Recall that an algebra (W,→,∗,1) of type (2,1,0) is aWajsberg algebra if it satisfies the following conditions, for all x, y, z∈W:
(W1) 1→x=x;
(W2) (x→y)→((y→z)→(x→z)) = 1;
(W3) (x→y)→y= (y→x)→x;
(W4) (x∗→y∗)→(y→x) = 1.
Following [26], if (W,→,∗,1) is a Wajsberg algebra, then (W,⊕,∗,0) is an M V-algebra, where forx, y∈W, x⊕y=x∗ →y.Conversely, if (W,⊕,∗,0) is an M V-algebra, then (W,→,∗,1) is a Wajsberg algebra, where for x, y ∈ W, x→y=x∗⊕y.
An order relation on a Wajsberg algebra W is given by:
x≤y iffx→y= 1.
Following Theorems 2.5, 2.6 and Corollary 2.7, if L is a semi-divisible residuated lattice and forx, y∈L we define
x∗⊕y∗ = (x∗)∗ →y∗ =x∗∗→y∗ =y→x∗∗∗ =y→x∗ =x→y∗, then M V(L) gets a structure of M V-algebra, and hence also a structure of Wajsberg algebra, and the order relation≤W on the Wajsberg algebraM V(L) is defined by x∗ ≤W y∗ iffx∗→y∗ = 1 iffx∗≤y∗, for all x, y∈L, where≤is the order of L.
Hence, the order relation on M V(L) coincides with the order relation on L.Also, it was mentioned in [25] that if L is a residuated lattice such that M V(L) is a Wajsberg algebra, then the implication→of this Wajsberg algebra coincides with the restriction of the operation → of L to M V(L). According to [25], the lattice operations least upper bound{x∗, y∗},denoted byx∗∨Wy∗, and greatest lower bound {x∗, y∗}, denoted by x∗ ∧W y∗, on the Wajsberg algebraM V(L) are defined via
x∗∨W y∗ = (x∗ →y∗)→y∗ and x∗∧W y∗= (x∗∗∨W y∗∗)∗.
In general for a residuated lattice L, the meet operation∧W on M V(L) needs not be a restriction of the meet operation ∧on Land, similarly, for the join operation.
We have, however, the following result:
Proposition 2.8 ([25]). If L is a semi-divisible residuated lattice, then
∧Wand∧ coincide on M V(L).
Lemma 2.9 ([7]). IfL is a semi-divisible residuated lattice, then for every x, y∈L we have:
(c42) [x∗(x∗→y∗)]∗ = (y∗∗→x∗∗)→x∗∗; (c43) x∗∗∨W y∗∗= (x∗∧y∗)∗ (c= (x31) ∨y)∗∗.
Proposition 2.10 ([7]). For a residuated lattice L, the following condi- tions are equivalent:
(i) L∈ RLsd;
(ii) (x∗∗→y∗∗)→y∗∗= (y∗∗→x∗∗)→x∗∗ for everyx, y∈L.
Remark 2.4.
1. Every divisible residuated lattice is semi-divisible, so, everyBL-algebra is semi-divisible.
2. Not every residuated lattice is semi-divisible. Consider, for example (see [25]) a fixed real number c,0 < c <1,and define the residuated lattice Lc= ([0,1],∨,∧,,→,0,1) such that for allx, y∈[0,1],
xy= 0 if x+y≤cand min{x, y} elsewhere, x→y= 1 if x≤y and max{c−x, y}elsewhere.
We have M V(Lc) = [0, c) ∪ {1}. Let x = 35c, y = 45c. Then x, y ∈ M V(Lc),(y → x) → x = 1, but (x → y) →y =y. Thus, the condition from Theorem 2.5 does not hold. So Lc is not semi-divisible. Evidently, each residuated lattice Lc is prelinear, therefore it is an M T L-algebra.
We deduce thatM T L-algebras are not semi-divisible in general.
3. There are residuated lattices that are semi-divisible but not divisible.
Consider, for example (see [25]) the residuated lattice LsD = ([0,1],∨,∧,,→,0,1) such that for allx, y∈[0,1],
xy= 0 if x, y∈[0,1
2] and min{x, y} elsewhere x→y= 1 if x≤y ; 1
2 ify < x≤ 1
2 and y if (y < x,1 2 < x).
We haveM V(LsD) ={0,12,1}and the condition from Theorem 2.5 holds, whence, LsD is a semi-divisible residuated lattice. LsD is not divisible.
Indeed, letx= 13, y= 12.Then 12(12 → 13) = 1212 = 06= 13 = min{13,12}.
We recall that if (Li,∨,∧,,→,0,1), i= 1,2 are two residuated lattices then, a mapf :L1 →L2 is calledmorphism of residuated lattices iff satisfies the following conditions, for every x, y∈L1 :
(mo1) f(0) = 0;
(mo2) f(1) = 1;
(mo3) f(x∧y) =f(x)∧f(y);
(mo4) f(x∨y) =f(x)∨f(y);
(mo5) f(x→y) =f(x)→f(y);
(mo6) f(xy) =f(x)f(y).
Remark 2.5.
1. Using the fact that in BL-algebras we have the equations (BL1) and x∨y = [(x → y) → y]∧[(y → x) → x] we deduce that if L1, L2 are BL-algebras, thenf :L1 →L2 is a morphism of residuated lattices ifff verifies (mo1),(mo5) and (mo6).
Analogously we deduce that:
2. If L1, L2 are divisible, then f : L1 → L2 is a morphism of residuated lattices ifff verifies the conditions (mo1),(mo4),(mo5) and (mo6).
3. If L1, L2 are M T L-algebras, then f :L1 → L2 is a morphism of residu- ated lattices ifff verifies (mo1)–(mo3) and (mo5)–(mo6).
RL becomes in a canonical way a category such that RLd,MT L,RLsd and BLare subcategories of RL.
Remark 2.6 ([1], p. 31). Since the categoriesMV andRLare equational, it follows that in these categories the monomorphisms are exactly the injective morphisms.
Definition 2.3 ([1], p. 27).A subcategory B of category A is said to be reflective if there is a functor R : A → B called reflector, such that for each A ∈ Ob(A), there exists a morphism ΦR(A) : A → R(A) of A with the following properties:
(i) If A0 ∈Ob(A) and f ∈HomA(A, A0),then ΦR(A0)◦f =R(f)◦ΦR(A), that is the diagram
A −→f A0
↓ΦR(A) ↓ΦR(A0)
R(A) R(f−→ R(A) 0) is commutative,
(ii) IfB ∈Ob(B),andf ∈HomA(A, B),then there exists a unique morphism f0 ∈HomB(R(A), B),such thatf0◦ΦR(A) =f, that is the diagram
A Φ−→R(A) R(A)
&
f
.
f0
B is commutative.
Theorem 2.11 ([1], p. 29). Suppose R :A → B is a reflector. Then R preserves inductive limits of partially ordered systems.
In what follows, byLwe denote the universe of a divisible residuated lat- tice. SinceLis in particular semi-divisible, by Corollary 2.7, (M V(L),⊕,∗,0) is anM V-algebra (hence, aBL-algebra), where forx, y∈M V(L), x∗=x→0∈ M V(L) and x⊕y =x∗ →y=x∗→y∗∗= (x∗y∗)∗ ∈M V(L).
The order onM V(L) is defined forx, y∈M V(L) byx≤W yiffx∗⊕y= 1 iffx∗∗→y= 1 iffx→y= 1 iffx≤y, so the order onM V(L) is the restriction of the order on LtoM V(L).
Also, if forx, y∈Lwe definex∗W y∗= [(x∗)∗⊕(y∗)∗]∗= (x∗∗⊕y∗∗)∗, then (M V(L),∨W,∧W =∧,→,W,0,1) is a BL-algebra (hence, a residuated lattice).
Proposition 2.12 ([7]). If x, y∈L, then(xy)∗∗=x∗∗W y∗∗. We denote R(L) = M V(L) and we define ΦR(L) : L → M V(L) by ΦR(L)(x) =x∗∗,for all x∈L.
Proposition 2.13 ([7]). IfL is a divisible residuated lattice, thenΦR(L) is a morphism of residuated lattices (between the residuated lattices (L,∨,∧,→ ,,0,1)and (M V(L),∨W,∧W =∧,→,W,0,1)).
As in the case of M V-algebras (see [3]), if L, L0 are divisible residuated lattices andf :L→L0 is a morphism of residuated lattices, then
R(f) :M V(L)→M V(L0)
defined by R(f)(x∗) =f(x∗) = (f(x))∗ for everyx∈Lis a morphism in MV.
Hence, the assignments L M V(L) = R(L) and f R(f) define a (covariant) functor R:RLd→ MV from the category of divisible residuated lattices to the category ofM V-algebras.
Theorem 2.14 ([7]). The category MV of M V-algebras is a reflective subcategory of the categoryRLd of divisible residuated lattices and the reflector R preserves monomorphisms.
2.3. BOOLEAN CENTER
Let (L,∨,∧,0,1) be a bounded lattice. Recall (see [1]) that an element a∈L is calledcomplemented if there is an element b∈L such that a∨b= 1 anda∧b= 0; if such an elementbexists, then it is called acomplement ofa.We will denote the set of all complemented elements in Lby B(L).Complements are generally not unique, unless the lattice is distributive.
In residuated lattices however, although the underlying lattices need not be distributive, the complements are unique (see [15]), which allows the nota- tion that follows for complements in residuated lattice.
Let L be a residuated lattice; we shall denote by B(L) the set of all complemented elements of its underlying bounded lattice (L,∨,∧,0,1). For everya∈B(L), we shall denote bya0 the unique complement of ainL.
For each e∈B(L), the following hold: e0 =e∗,e∗∗=e,ex=e∧x, for everyx∈L,and hence,e2 =eande→x=e∗∨x=e0∨xfor everyx∈L(see [15] and (c51) below). Now it is easy to see that the setB(L) is the universe of a Boolean algebra which is also a subalgebra of the residuated lattice L, called theBoolean center of L (see [15]).
In the residuated latticeB(L),=∧and→ coincides with the Boolean algebra implication, as mentioned above, which shows that B(L) is prelinear and divisible, that is,B(L) is a BL-algebra.
Proposition 2.15 ([5]). If L is a residuated lattice, then for e∈ L the following are equivalent:
(i) e∈B(L);
(ii) e∨e∗ = 1.
Theorem 2.16 ([8]). For every element e in an M V-algebra A, the fol- lowing conditions are equivalent:
(i) e∈B(A);
(ii) e∨e∗ = 1;
(iii) e∧e∗ = 0;
(iv) e⊕e=e;
(v) ee=e.
Proposition 2.17 ([5]). Let L be a residuated lattice. For e ∈ L we consider the following assertions:
(i) e∈B(L);
(ii) e2=eand e=e∗∗; (iii) e2=eand e∗ →e=e;
(iv) (e→x)→e=e, for every x∈L;
(v) e∧e∗ = 0.
Then (i) ⇒ (ii),(iii),(iv) and (v) but (ii) ; (i),(iii) ; (i),(iv) ; (i),(v);(i).
Remark 2.7 ([7]). IfLis a divisible residuated lattice, then the assertions (i), (ii), (iii) and (iv) from Proposition 2.17 are equivalent.
Theorem 2.18 ([7]). Let L be a residuated lattice. For all e ∈ L the following assertions are equivalent:
(i) e∈B(L);
(ii) e=e∗∗, ee=eand e∗(e∗→e) = 0.
Proposition 2.19. If L is a divisible residuated lattice, then B(L) =B(M V(L)).
Proof. By the above, for every e ∈ B(L), we have: e = e∗∗ ∈ M V(L) and ee = e, hence B(L) ⊆ M V(L)∩ {a ∈ L : aa = a}. Remark 2.4 and Corollary 2.7 ensure us that M V(L) is an M V-algebra, hence, by Theo- rem 2.16, M V(L)∩ {a ∈ L : aa = a} = B(M V(L)). We have obtained:
B(L)⊆B(M V(L))⊆B(L), hence B(L) =B(M V(L)). For the last inclusion before, we have used the fact that 0,1∈M V(L) and the order onM V(L) co- incides with the restriction of the order onL toM V(L), which shows that the underlying bounded lattice ofM V(L) is a bounded sublattice of the underlying bounded lattice of L.
Lemma 2.20 ([7]). Let L be a residuated lattice. If x, y ∈ L and e, f ∈ B(L), then:
(c42) x(x→e) =x∧e, e(e→x) =e∧x;
(c43) e∨(xy) = (e∨x)(e∨y);
(c44) e∧(xy) = (e∧x)(e∧y);
(c45) e(x→y) =e[(ex)→(ey)];
(c46) x(e→f) =x[(xe)→(xf)];
(c47) e→(x→y) = (e→x)→(e→y);
(c48) e→(e→x) =e→x;
(c49) (e→x)→e=e;
(c50) (e→x)→x≤(x→e)→e;
(c51) e∗→x= (e→x)→x=e∨x;
(c52) e∨(x→y) = (e∨x)→(e∨y);
(c53) (e∨x)→(f∨x) = (e→f)∨x;
(c54) x→(e→f) = (x→e)→(x→f);
(c55) e∧(x∨y) = (e∧x)∨(e∧y);
(c56) x∧(e∨f) = (x∧e)∨(x∧f);
(c57) If e, f ≤x,then e(x→f) =f(x→e);
(c58) (e→x)∨(x→e) = 1;
(c59) e∨x= [(e→x)→x]∧[(x→e)→e].
Proposition 2.21. Let L be a residuated lattice. Ifx∈Land e∈B(L), then:
(c60) (e∧x)∗ =e∗∨x∗.
Proof. By (c7),we have (e∧x)∗≥e∗∨x∗. Conversely,
e→x= 1∧(e→x)(c= (e2) →e)∧(e→x)(c=7)e→(e∧x)
(c26)
≤ (e∧x)∗ →e∗ and x→e=x→(x∧e)
(c26)
≤ (e∧x)∗ →x∗, so (e→x)(e∧x)∗ ≤e∗ and (x→e)(e∧x)∗ ≤x∗, by the law of residuation.
It follows that
(e∧x)∗ = 1(e∧x)∗(c= [(e58) →x)∨(x→e)](e∧x)∗
(c21)
= [(e→x)(e∧x)∗]∨[(x→e)(e∧x)∗]≤e∗∨x∗. We conclude that (e∧x)∗ =e∗∨x∗.
Lemma 2.22. Let L be a divisible residuated lattice and a, b, x∈ L such that a, b≤x. Then
(c61) a(x→b) =b(x→a).
Proof. We have
a(x→b) = (x∧a)(x→b) = [x(x→a)](x→b)
= [x(x→b)](x→a) = (x∧b)(x→a) =b(x→a).
3. MULTIPLIERS ON A DIVISIBLE RESIDUATED LATTICE
In the rest of this paper, byLwe denote an arbitrary divisible residuated lattice, unless mentioned otherwise.
We denote by Id(L) the set of all ideals of the lattice Land by I(L) the set of all non-empty decreasing subsets ofL, that is,
I(L) ={I ⊆L:I 6=∅and, if x, y∈L, x≤y and y∈I,thenx∈I}.
Remark 3.1.Clearly, Id(L) ⊆ I(L) and if I1, I2 ∈ I(L), then I1∩I2 ∈ I(L).Also, if I ∈I(L),then 0∈I.
Definition 3.1. By partial multiplier on L we mean a map f : I → L, whereI ∈I(L),which verifies the next conditions:
(m1) f(ex) =ef(x),for every e∈B(L) and x∈I; (m2) f(x)≤x, for everyx∈I;
(m3) Ife∈I∩B(L),then f(e)∈B(L);
(m4) x∧f(e) =e∧f(x),for everye∈I∩B(L) and x∈I.
By dom(f) ∈I(L) we denote the domain off; ifdom(f) = L,then f is called total.
To simplify the language, we will use multiplier instead of partial multi- plier using total to indicate that the domain of a certain multiplier isL.
Example 3.1.
1. The map 0 : L → L defined by 0(x) = 0, for every x ∈ L is a total multiplier on L; indeed if x ∈ L and e ∈ B(L), then 0(ex) = 0 = e0 =e0(x) and 0(x)≤x.
Clearly, if e∈L∩B(L) = B(L),then 0(e) = 0∈ B(L) and for x ∈L, x∧0(e) =e∧0(x) = 0.
2. The map 1:L→L defined by 1(x) =x, for every x∈L is also a total multiplier onL; indeed if x∈L and e∈B(L), then 1(ex) =ex= e1(x) and1(x) =x≤x.
The conditions (m3)–(m4) are obviously verified.
3. For arbitrary a ∈ L and I ∈ I(L), we shall denote by fa : I → L the map that takes every x ∈ L to fa(x) = a∧x. If a∈ B(L), then fa is a multiplier of L, calledprincipal multiplier. Indeed, consider a∈B(L);
then forx∈I ande∈B(L), we havefa(ex) =a∧(ex) =a∧(e∧x) = e∧(a∧x) =e(a∧x) =efa(x) and clearlyfa(x)≤x.
Also, ife∈I∩B(L),thenfa(e) =e∧a∈B(L) andx∧(a∧e) =e∧(a∧x), for everyx∈I.
Remark 3.2. The condition (m4) is not a consequence of (m1)–(m3).As an example, let I ∈I(L) andf :I → L, f(x) =x∧x∗ for every x ∈I. Then f satisfies (m1)–(m3), but it does not satisfy (m4). Indeed, for x ∈ I and e∈B(L), we have
f(ex) = (ex)∧(ex)∗ = (e∧x)∧(e∧x)∗
(c60)
= (e∧x)∧(e∗∨x∗) =x∧[e∧(e∗∨x∗)]
(c55)
= x∧[(e∧e∗)∨(e∧x∗)] =x∧[0∨(e∧x∗)]
= x∧(e∧x∗) =e∧(x∧x∗) =e(x∧x∗) =ef(x) and clearly f(x)≤x.Also, ife∈I∩B(L),thenf(e) =e∧e∗ = 0∈B(L) but
ife∈I∩B(L) and x∈I, then
x∧f(e) =x∧0 = 0 is not necessarily equal to e∧(x∧x∗) =e∧f(x).
Indeed, in the following example of divisible residuated lattice L, there exist I ∈ I(L), x ∈ I and e ∈ B(L) such that x∧f(e) 6= e∧f(x) (that is, e∧(x∧x∗)6= 0).
Let L={0, α,1} be the three—element chain and the operations and
→ given by the tables:
0 α 1
0 0 0 0
α 0 0 α
1 0 α 1
→ 0 α 1
0 1 1 1
α α 1 1
1 0 α 1
Then it is routine to prove that (L,∨,∧,,→,0,1) is a divisible residu- ated lattice with B(L) ={0,1}.
If we consider I =L,e= 1∈I∩B(L) and x=α∈I, then e∧(x∧x∗) =α∧α∗ =α∧α=α6= 0.
Remark 3.3. In general, if we considera∈L andI ∈I(L), thenfa:I → L verifies (m1),(m2) and (m4) but it does not verify (m3).
If dom(fa) =L,we denote fa byfa ; clearly, f0=0 andf1 =1.
For I ∈I(L),we denote
M(I, L) ={f :I →L|f is a multiplier on L}
and
M(L) = [
I∈I(L)
M(I, L).
If necessary, we denote M(I, L) by MRLd(I, L) to indicate that we work in divisible residuated lattices; for the case ofM V-algebras we denoteM(I, L) by MMV(I, L).
Remark 3.4.From Proposition 2.19 we deduce that for every I ∈ I(L) the algebra of multipliers MRLd(I, L) for a divisible residuated lattice L is in fact a generalization of the algebra of multipliersMMV(I, L) for M V-algebras (see [3], Definition 3.1). Also, we deduce that if L is an M V-algebra (that is L=M V(L)),thenMRLd(I, L) =MMV(I, L) for everyI ∈I(L).
Definition 3.2.If I1, I2 ∈ I(L) and fi ∈ M(Ii, L), i = 1,2, we define f1∧f2, f1∨f2, f1f2, f1 →f2:I1∩I2 →Lby
(f1∧f2)(x) =f1(x)∧f2(x),
(f1∨f2)(x) =f1(x)∨f2(x),
(f1f2)(x) =f1(x)[x→f2(x)](c=61)f2(x)[x→f1(x)] = (f2f1)(x), (f1 →f2)(x) =x[f1(x)→f2(x)].
for everyx∈I1∩I2.
In the following four lemmas, we shall use the notations in Definition 3.2.
As in [4] forBL-algebras, we immediately obtain the following two results:
Lemma 3.1. f1∧f2∈M(I1∩I2, L).
Lemma 3.2. f1∨f2∈M(I1∩I2, L).
Lemma 3.3. f1f2 ∈M(I1∩I2, L).
Proof. If x∈I1∩ I2 and e∈B(L),then
(f1f2)(ex) = f1(ex)[(ex)→f2(ex)]
= [ef1(x)][(ex)→(ef2(x))]
= f1(x)[e((ex)→(ef2(x)))]
(c45)
= f1(x)[e(x→f2(x))]
= e[f1(x)(x→f2(x))] =e(f1f2)(x).
Clearly, (f1f2)(x) =f1(x)[x→f2(x)]≤f1(x)≤x,for everyx∈I1∩ I2 and if e∈I1∩I2∩B(L),then
(f1f2)(e) =f1(e)[e→f2(e)] =f1(e)∧(e∗∨f2(e))∈B(L).
For e∈I1∩I2∩B(L) and x∈I1∩ I2 we have:
x∧(f1f2)(e) = x∧[f1(e)(e→f2(e))] =x[f1(e)(e→f2(e))]
= f1(e)[x(e→f2(e))]
(c46)
= f1(e)[x[(xe)→(xf2(e))]]
= (f1(e)x)[(xe)→(xf2(e))]
= (ef1(x))[(ex)→(ef2(x))]
= f1(x)[e[(ex)→(ef2(x))]]
(c45)
= f1(x)[e[x→f2(x)]]
= e[f1(x)(x→f2(x))] =e(f1f2)(x)
= e∧(f1f2)(x), hence,
x∧(f1f2)(e) =e∧(f1f2)(x).
Therefore f1f2∈M(I1∩I2, L).
Lemma 3.4. f1 →f2 ∈M(I1∩I2, L).
Proof. If x∈I1∩ I2 and e∈B(L),then
(f1 →f2)(ex) = (ex)[f1(ex)→f2(ex)]
= (ex)[(ef1(x))→(ef2(x))]
= x[e((ef1(x))→(ef2(x)))]
(c45)
= x[e(f1(x)→f2(x))]
= e[x(f1(x)→f2(x))] =e(f1 →f2)(x).
Clearly, (f1 → f2)(x) = x[f1(x) → f2(x)] ≤ x, for every x ∈ I1∩ I2
and if e∈I1∩I2∩B(L),then
(f1 →f2)(e) = e[f1(e)→f2(e)] =e[(f1(e))∗∨f2(e)]
= e∧[(f1(e))∗∨f2(e)]∈B(A).
For e∈I1∩I2∩B(A) and x∈I1∩ I2 we have:
e∧(f1→f2)(x) = e∧[x(f1(x)→f2(x))]
= (ex)[f1(x)→f2(x)]
= x[e(f1(x)→f2(x))]
(c45)
= x[e((ef1(x))→(ef2(x)))]
= x[e((xf1(e))→(xf2(e)))]
= e[x((xf1(e))→(xf2(e)))]
(c46)
= e[x(f1(e)→f2(e))]
= x[e(f1(e)→f2(e))] =x(f1 →f2)(e)
= x∧(f1→f2)(e) hence,
x∧(f1 →f2)(e) =e∧(f1 →f2)(x).
Therefore f1→f2 ∈M(I1∩I2, L).
We recall the notations 0 and 1 from the examples above: 0,1:L→L, for all x∈L,0(x) = 0 and1(x) =x. divisible residuated lattice.
Lemma3.5. The operations defined in Definition 3.2satisfy the following properties:
(i) f1(f2f3) = (f1f2)f3 for everyf1, f2, f3 ∈M(L);
(ii) f1f2 =f2f1, for every f1, f2 ∈M(L);
(iii) f1=1f =f for everyf ∈M(L);
(iv) f1(x) ≤ (f2 → f3)(x) iff (f1 f2)(x) ≤ f3(x) for every fi ∈ M(Ii, L), where Ii ∈I(L), for each i= 1,2,3, and for all x∈I1∩I2∩I3;
(v) f1(f1→f2) =f1∧f2 for allf1, f2∈M(L).
Proof.
(i) By Lemma 3.3, f g ∈ M(L), for all f, g ∈ M(L). Let fi ∈ M(Ii, L) whereIi∈I(L), i= 1,2,3.
Thus, forx∈I1∩I2∩I3 we have
[f1(f2f3)](x) = ((f2f3)(x))(x→f1(x))
= [f2(x)(x→f3(x))](x→f1(x))
= f2(x)[(x→f3(x))(x→f1(x))]
= f2(x)[(x→f1(x))(x→f3(x))]
= [f2(x)(x→f1(x))](x→f3(x))
= = ((f1f2)(x))(x→f3(x))
= [(f1f2)f3](x), that is the operation is associative.
(ii) By definition
(f1f2)(x) = f1(x)[x→f2(x)](c=61)f2(x)[x→f1(x)]
= (f2f1)(x), that is the operation is commutative.
(iii) Letf ∈M(I, L) withI ∈I(L).Ifx∈I,then
(f 1)(x) = f(x)(x→1(x)) =f(x)(x→x)
= f(x)1 =f(x),
hence,f1=f, and by the commutativity ofwe have 1f =f 1=f.
(iv) Letx∈I1∩I2∩I3, arbitrary. Then:
f1(x)≤(f2 →f3)(x)⇔f1(x)≤x[f2(x)→f3(x)].
So, by (c11), iff1(x)≤(f2→f3)(x), then
f1(x)[x→f2(x)]≤x(x→f2(x))(f2(x)→f3(x))⇒ f1(x)[x→f2(x)]≤(x∧f2(x))(f2(x)→f3(x))⇒ f1(x)[x→f2(x)]≤f2(x)(f2(x)→f3(x))⇒
f1(x)[x→f2(x)]≤f2(x)∧f3(x)≤f3(x)⇒ (f1f2)(x)≤f3(x).
Conversely, if (f1f2)(x)≤f3(x) then
f1(x)[x→f2(x)]≤f3(x), for allx∈I1∩I2∩I3. Since,
f2(x)[x→f1(x)]≤f1(x)[x→f2(x)] (see Lemma 2.22),⇒ x→f1(x)≤f2(x)→[f1(x)(x→f2(x))]
= f2(x)→(f1f2)(x)
(c9)
≤ f2(x)→f3(x)
(c11)
⇒ x(x→f1(x))≤x(f2(x)→f3(x))
⇒x∧f1(x)≤x(f2(x)→f3(x))⇒f1(x)≤(f2 →f3)(x).
(v) Letfi ∈M(Ii, L) whereIi ∈I(L), i= 1,2.
Thus, forx∈I1∩I2 we have
[f1(f1 →f2)](x) = [(f1 →f2)(x)][x→f1(x)]
= x[f1(x)→f2(x)][x→f1(x)]
= (x[x→f1(x)])[f1(x)→f2(x)]
= [x∧f1(x)][f1(x)→f2(x)]
= f1(x)[f1(x)→f2(x)] =f1(x)∧f2(x)
= (f1∧f2)(x).
So,
f1∧f2 =f1(f1 →f2)
Corollary 3.6. For every I ∈I(L), (M(I, L),∨,∧,,→,0|I,1|I) is a divisible residuated lattice.
Proof. Clearly, (M(I, L),∨,∧,0|I,1|I) is a bounded lattice.
(i), (ii) and (iii) from Lemma 3.11 applied to multipliers from M(I, L) show that (M(I, L),,1|I) is a commutative monoid. In the same way, (iv) from Lemma 3.11 proves the law of residuation for M(I, L), while (v) from Lemma 3.11 proves thatM(I, L) satisfies (BL1).
We also have for divisible residuated lattices the next analogous defini- tions, results and remarks as in [4] for BL-algebras.
Lemma3.7. The mapvL:B(L)→M(L)defined byvL(a) =fafor every a∈B(L),is an injective map which verifies the following conditions:
(i) vL(0) =0 and vL(1) =1;
(ii) vL(a∨b) =vL(a)∨vL(b) for everya, b∈B(L);
(iii) vL(a→b) =vL(a)→vL(b) for every a, b∈B(L);
(iv) vL(ab) =vL(a)vL(b) for every a, b∈B(L).
Proof. We begin by proving the injectivity of vL. Ifa, b∈B(L) and vL(a) =vL(b), thenfa=fb hence a∧x=b∧x,
for all x∈L. In particular for x= 1 we obtain a=b.
(i) ClearlyvL(0) =f0 =0 and vL(1) =f1=1.
(ii) Ifa, b∈B(L), thenvL(a∨b) =vL(a)∨vL(b) iff (a∨b)∧x= (a∧x)∨(b∧x), for all x∈L, which is a consequence of (c56).
(iii) Ifa, b∈B(L), then
vL(a→b) =vL(a)→vL(b)iff (a→b)∧x=x[(a∧x)→(b∧x)]
for all x∈L. We have:
x[(a∧x)→(b∧x)](c=7)x[((a∧x)→b)∧((a∧x)→x)](c=2) x[((a∧x)→b)∧1] =x[(a∧x)→b] =x[(ax)→b].
On the other hand,
(a→b)∧x=x[x→(a→b)](c=14)x[(xa)→b], hence,vL(a→b) =vL(a)→vL(b).
(iv) Ifa, b∈B(L) then
vL(ab) =vL(a)vL(b) iff (ab)∧x= (a∧x)[x→(b∧x)]
for all x∈L. We have:
(a∧x)[x→(b∧x)](c= (a7) ∧x)[(x→b)∧(x→x)] = (a∧x)(x→b) =ax(x→b) =a(x∧b) = a(xb) = (ab)∧x,
hence,vL(ab) =vL(a)vL(b).
Definition 3.3. A nonempty setI ⊆Lis calledregularif for everyx, y∈L such that x∧e=y∧efor every e∈I∩B(L),it follows that x=y.
For exampleLis a regular subset of L(since ifx, y∈Landx∧e=y∧e for everye∈L∩B(L) =B(L),then fore= 1 we obtain x∧1 =y∧1, that is x=y).
More generally, every subset of Lwhich contains 1 is regular.
We denote
R(L) ={I ⊆L:I is a regular subset of L}.
Remark 3.5. For every nonempty subset I of L, the conditionI ∈ R(L) is equivalent with the condition: for every x, y ∈ L, if fx|I∩B(L) =fy|I∩B(L), thenx=y.
Lemma 3.8. If I1, I2 ∈I(L)∩R(L), thenI1∩I2∈I(L)∩R(L).
Proof. Clearly I1 ∩I2 ∈ I(L). Let x, y ∈ L such that x∧e = y∧e for all e ∈ I1∩I2 ∩B(L). We will prove that x = y. For this, if we consider e1∈I1∩B(L) ande2 ∈I2∩B(L) it follows thate1∧e2 ∈I1∩I2∩B(L), hence x∧e1∧e2 =y∧e1∧e2⇒(x∧e1)∧e2= (y∧e1)∧e2 ⇒x∧e1=y∧e1 ⇒x=y.
We denote L = M V(L) and for I ∈ I(L), I = I ∩L. Then MRLd(I, L) becomes a divisible residuated lattice andMMV(I, L) becomes anM V-algebra.
Remark 3.6. We remark that by Definition 3.2, forf, g∈M V(MRLd(I, L)), fg:I →Lis defined by (fg)(x) =f(x)(x→g(x)) for everyx∈I. If we definedf⊕g:I →Lbyf⊕g= (f∗g∗)∗, then (M V(MRLd(I, L)),⊕) becomes an M V-algebra (see [22]). In the case of MMV(I, L), for f, g ∈ MMV(I, L), f ⊕Lg :I → L is defined by (f ⊕Lg)(x) = (f(x)⊕g(x))∧x for every x∈ I (see [22], Proposition 6.9).
Proposition 3.9. If I ∈I(L)∩R(L), then (i) I ∈I(L)∩R(L),
(ii) The map ϕI :M V(MRLd(I, L))→ MMV(I, L), ϕI(f) =f|I is an injec- tive morphism in MV.
Proof. (i). Clearly I ∈ I(L); to prove that I ∈ R(L) let x, y ∈ L such that x∧e=y∧e for every e∈ I∩B(L) =I ∩L∩B(L) =I ∩B(L) (since B(L)⊆L).SinceI ∈R(L), it follows that x=y, that is,I ∈R(L).
(ii) By Definition 3.2, for all f ∈MRLd(I, L), f∗ =f →0∈MRLd(I, L) is defined for x∈I by:
f∗(x) = (f →0)(x) =x(f(x)→0(x)) =x(f(x)→0) =x(f(x))∗, thus,
f∗∗(x) =x[x(f(x))∗]∗ (c=30)x(x→(f(x))∗∗)(BL=1)x∧(f(x))∗∗=x∧f∗∗(x).
Now, let f ∈M V(MRLd(I, L)). Then from f∗∗=f, we deduce that, for all x∈I, x∧(f(x))∗∗=x∧f∗∗(x) =x∧f(x) =f(x).
Ifx∈I =I∩L,thenx=x∗∗,so fromx∧(f(x))∗∗=f(x) we deduce that x∗∗∧(f(x))∗∗=f(x) (c⇒39)(x∧f(x))∗∗=f(x) ⇒(f(x))∗∗=f(x)⇒f(x)∈L, hence ϕI is well defined.
To prove the injectivity of ϕI , let f, g ∈ M V(MRLd(I, L)) such that ϕI(f) =ϕI(g)⇔ f|I∩L=g|I∩L.
Letx∈I and e∈I∩B(L).SinceB(L)⊆L,we deduce thatI∩B(L)⊆ I ∩L,hence, f(e) =g(e). Then x∧f(e) =x∧g(e) (m⇒4) f(x)∧e= g(x)∧e.
Since I ∈ R(L) we deduce that f(x) = g(x), hence, f = g, that is ϕI is an injective map. We have ϕI(0) =0|I.
To prove that ϕI is a morphism in MV let f, g∈M V(MRLd(I, L)) and see Remark 2.2 and 2.3.
Then ϕI(f ⊕g) = (f ⊕g)|I , hence forx∈ I,we have
(ϕI(f⊕g))(x) = (f ⊕g)(x) = ((f∗)∗⊕(g∗)∗)(x) = (f∗g∗)∗(x)
= x[(f∗g∗)(x)]∗ =x[f∗(x)(x→g∗(x))]∗
= x[(x(f(x))∗)(x→(x(g(x))∗))]∗
= x[(f(x))∗(x∧(x(g(x))∗))]∗
= x[(f(x))∗x(g(x))∗]∗
= x[x(f(x))∗(g(x))∗]∗
(c30)
= x[x→((f(x))∗(g(x))∗)∗]
= x∧[(f(x))∗(g(x))∗]∗
= ((f(x))∗∗⊕(g(x))∗∗)∧x
= (f(x)⊕g(x))∧x= (f|I⊕
Lg|I)(x)
= (ϕI(f)⊕LϕI(g))(x), hence,
ϕI(f⊕g) =ϕI(f)⊕LϕI(g).
Also, for any x∈I∩L=I,ϕI(f∗)(x) =f∗(x) = (ϕI(f))∗(x), hence ϕI(f∗) = (ϕI(f))∗,
that is ϕI is an injective morphism in MV.
We denote Mr(L) ={f ∈M(L) :dom(f)∈I(L)∩R(L)}.
Remark 3.7.From Lemma 3.8, we deduce that if f1, f2 ∈Mr(L), then f1∨f2, f1∧f2, f1f2, f1→f2 ∈Mr(L).
Definition 3.4.Given two multipliers f1, f2 on L,we say thatf2 extends f1 ifdom(f1)⊆dom(f2) andf2|dom(f1)=f1; we writef1vf2 iff2 extendsf1. A multiplier f is calledmaximal if f can not be extended to a strictly larger domain.
As in [4] forBL-algebras, we have:
Lemma 3.10.
(i) If f1, f2 ∈M(L), f ∈Mr(L) andf vf1, f vf2,thenf1 and f2 coincide ondom(f1)∩dom(f2),
(ii) Every multiplier f ∈ Mr(L) can be extended to a maximal multiplier.
More precisely, each principal multiplierfawitha∈B(L)anddom(fa)∈ I(L)∩R(L) can be uniquely extended to the total multiplier fa and each non-principal multiplier can be extended to a maximal non-principal one.
On Mr(L) we consider the relation ρL defined by
(f1, f2)∈ρL ifff1 and f2 coincide on the intersection of their domains.
Lemma 3.11. ρL is an equivalence on the set Mr(L) which is compatible with the operations ∨,∧,and →.
Proof.The reflexivity and the symmetry ofρLare immediate; to prove the transitivity of ρL let (f1, f2),(f2, f3) ∈ ρL. Therefore f1, f2 and respectively, f2, f3 coincide on the intersection of their domains. If by contrary, there exists x0 ∈ dom(f1) ∩dom(f3) such that f1(x0) 6= f3(x0), then, since dom(f2) ∈ R(L), there exists e ∈ dom(f2)∩B(L) such that e∧f1(x0) 6= e∧f3(x0) ⇔ ef1(x0)6=ef3(x0)⇔f1(ex0)6=f3(ex0) which is contradictory, since ex0 ∈dom(f1)∩dom(f2)∩dom(f3).
To prove the compatibility of ρL with the operations ∧,∨, and → on Mr(L), let (f1, f2),(g1, g2) ∈ ρL. So, we have f1, f2 and respectively, g1, g2 agree on the intersection of their domains.
Now to prove that
(f1∧g1, f2∧g2),(f1∨g1, f2∨g2),(f1g1, f2g2),(f1→g1, f2 →g2)∈ρL, let x ∈ dom(f1)∩dom(f2) ∩dom(g1)∩dom(g2). Then f1(x) = f2(x) and g1(x) =g2(x),hence,
(f1∧g1)(x) =f1(x)∧g1(x) =f2(x)∧g2(x) = (f2∧g2)(x), (f1∨g1)(x) =f1(x)∨g1(x) =f2(x)∨g2(x) = (f2∨g2)(x), (f1g1)(x) =f1(x)(x→g1(x)) =f2(x)(x→g2(x)) = (f2g2)(x) and
(f1→g1)(x) =x[f1(x)→g1(x)] =x[f2(x)→g2(x)] = (f2→g2)(x),
that is, the members of each of the pairs (f1∧g1, f2∧g2), (f1∨g1, f2∨g2), (f1g1, f2g2), (f1→g1, f2 →g2)
coincide on the intersection of their domains, hence ρL is compatible with the operations ∧,∨,and →.
For f ∈ Mr(L) with I = dom(f) ∈ I(L) ∩R(L), we denote by [f, I] the congruence class of f modulo ρL and L00 = Mr(L)/ρL. On L00 we define the order relation [f, I] ≤ [g, J] iff f(x) ≤ g(x) for every x ∈ I ∩J (when I =dom(f) and J =dom(g)).
It is routine to prove the following result:
Lemma 3.12. (L00,≤) is a bounded lattice, where [f, I]∧[g, J] = [f ∧ g, I∩J] and [f, I]∨[g, J] = [f ∨g, I∩J] (where f ∧g and f ∨g are defined in Definition 3.2) with first element [0, L] and last element [1, L].
For [f, I],[g, J]∈L00 we define [f, I][g, J] = [f g, I∩J] and [f, I]→ [g, J] = [f →g, I∩J] (where fgand f →g are defined in Definition 3.2).
Proposition 3.13. (L00,∧,∨,,→,[0, L],[1, L]) is a divisible residuated lattice.
Proof. We verify the axioms of a divisible residuated lattice.
(LR1) Follows from Lemma 3.12;
(LR2) Follows from Lemma 3.5, (i)–(iii);
(LR3) Follows from Lemma 3.5, (iv).
(BL1) Letfi∈M(Ii, L) whereIi∈I(L), i= 1,2.
Thus, from Lemma 3.11, (v) we deduce that f1(f1 →f2) =f1∧f2. So, [f1∧f2, I1∩I2] = [f1(f1 →f2), I1∩I2], hence
[f1, I1]∧[f2, I2] = [f1, I1]([f1, I1]→[f2, I2]), that is L00 is divisible.
Proposition 3.14. If we denote by F = I(L)∩R(L) and consider the partially ordered system {δI,J}I,J∈F,I⊆J, where for I, J ∈ F , I ⊆ J, δI,J : M(J, L)→M(I, L) is defined by δI,J(f) =f|I, then by the construction ofL00 above we deduce that
L00= lim−−→
I∈F
M(I, L) (in the categoryRLd).
